Projections In Banach Spaces Research Articles

Approximating properties of metric and generalized metric projections in uniformly convex and uniformly smooth Banach spaces

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.

Continuity of generalized metric projections in Banach spaces

Let $$\mathcal {P}$$ be the family of all proximinal subsets of a Banach space X. Let $$P:(X,\mathcal {P})\rightarrow 2^{X}$$ be the generalized metric projection defined as $$P(x,A)=P_{A}(x)=\{a\in A:\Vert x-a\Vert =d(x,A)\}$$ for any $$(x,A)\in (X,\mathcal {P})$$ , where $$P_{A}$$ is the usual metric projection on X. The mapping P is said to be (resp. weakly) upper semi-continuous at $$(x,A)\in (X,\mathcal {P})$$ in the Hausdorff sense if, for any (resp. weakly) open set $$W\supset P_A(x)$$ , any $$\{x_n\}_{n=1}^{\infty }\subset X$$ with $$x_n\rightarrow x$$ and any sequence $$\{A_n\}_{n=1}^{\infty }\subset \mathcal {P}$$ with $$A_n\xrightarrow {H}A$$ , there exists a $$N\in \mathbb {N}$$ such that $$P_{A_n}(x_n)\subset W$$ for any $$n>N$$ . In this paper, the continuity of $$P:(X,\mathcal {P})\rightarrow P_{A}(x)$$ are discussed. We prove that: (1) if X is a nearly strongly convex (resp. nearly very convex) space, then for any $$x\in X$$ and any convex subset $$A\in \mathcal {P}$$ the mapping $$P:(X,\mathcal {P})\rightarrow 2^{X},P(x,A)=P_{A}(x)$$ is (resp. weakly) upper semi-continuous at (x, A) in the Hausdorff sense; (2) if X has the property S (resp. property WS), then for any $$x^*\in X^*$$ and any convex subset $$A^*\in \mathcal {P}^*$$ with non-empty $$w^*$$ -interior points the mapping $$P:(X^*,\mathcal {P}^*)\rightarrow 2^{X^*},P(x^*,A^*)=P_{A^*}(x^*)$$ is (resp. weakly) upper semi-continuous at $$(x^*, A^*)$$ in the Hausdorff sense, where $$\mathcal {P}^*$$ stands for the family of all proximinal subsets of the dual space $$X^*$$ . Our results are generalizations of some known results concerning the continuity of the classical metric projection.

Kazhdan projections, random walks and ergodic theorems

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

Averaged projections, angles between groups and strengthening of Banach property (T)

Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties and Banach expanders. In this paper, we introduce a new strengthening of Banach property (T), called “robust Banach property (T)”, which is weaker than strong Banach property (T), but is still strong enough to ensure similar applications. Using the method of averaged projections in Banach spaces and introducing a new notion of angles between projections, we establish a criterion for robust Banach property (T) and show several examples of groups in which this criterion is fulfilled. We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.

The Problem of Image Recovery by the Metric Projections in Banach Spaces

We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.

On the uniqueness of minimal projections in Banach spaces

Let \(X\) be a uniformly convex Banach space with a continuous semi-inner product. We investigate the relation of orthogonality in \(X\) and generalized projections acting on \(X\). We prove uniqueness of orthogonal and co-orthogonal projections.

Nonlinear Operators of Monotone Type and Convergence Theorems with Equilibrium Problems in Banach Spaces

Our purpose in this paper is first to discuss nonlinear operators and nonlinear projections in Banach spaces which are related to the resolvents of $m$-accretive operators and maximal monotone operators. Some of these operators in Banach spaces are new. Next, we discuss some properties for such nonlinear operators and nonlinear projections in Banach spaces. Further, using these properties, we prove strong convergence theorems by hybrid methods for nonlinear operators with equilibrium problems in Banach spaces.

Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

According to the von Neumann–Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinati

On the non-uniqueness of minimal projection in [formula omitted] spaces

The main objective of this note is to exhibit a simple example of subspaces U ⊂ L p ( μ ) ( p ≠ 2 ) that admit two different projections with minimal norm. While for p = 1 , ∞ , such subspaces are well-known [W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1 < p < ∞ their existence was open.

Compositions of Projections in Banach Spaces and Relations Between Approximation Properties

A necessary and sucient condition for existence of a Banach space with a finite dimensional decomposition but without the -property in terms of norms of com- positions of projections is found.

A new projection and convergence theorems for the projections in Banach spaces

In this paper, we introduce a new projection in a Banach space and show an example of the projections. Then, we study the Mosco convergence of a sequence of nonempty sets concerning the projections in a Banach space.

Estimates for projections in Banach spaces and existence of direct complements

Let $W$ and $L$ be complementary subspaces of a Banach space $X$ and let $P(W,L)$ denote the projection on $W$ along $L$. We obtain a sufficient condition for a subspace $M$ of $X$ to be complementary to $W$ and we derive estimates for the norm of $P(W,L)

Sequential closedness of Boolean algebras of projections in Banach spaces

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950’s. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a σ-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).

The strong closure of Boolean algebras of projections in Banach spaces

AbstractThis note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

THE BEST APPROXIMATION BY PROJECTIONS IN BANACH SPACES

We consider the best approximation by projections in Banach spaces under certain suitable conditions. Furthermore, applications are discussed for multiplier operators and convolution type operators associated with strongly continuous families of bounded linear operators as well as for homogeneous Banach spaces which include the classical function spaces, as particular cases.

Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

The intimate connection between spectral measures and σ-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s, [“Linear Operators III: Spectral Operators,” Wiley-Interscience, New York, 1971; Chap. XVII]. It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just σ-complete. This makes it desirable to have available criteria (not just in Banach spaces but also in the nonnormable setting) which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal.

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

Characterizations of metric projections in Banach spaces and applications

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi‐inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

Convergence of orthogonal series of projections in Banach spaces

For a sequence $(A_j)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n = ∑^n_{j=1} A_j$ in a "strong" sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In th

On the uniform continuity of metric projections in Banach spaces

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smooth Banach space. It is proved that the metric projection PM of X onto M is uniformly continuous on every bounded subset of X. Moreover, a global and explicit estimate on the modulus of continuity of the metric projection is obtained.